Generic family with robustly infinitely many sinks

Abstract

We show, for every $r>d\ge 0$ or $r=d\ge 2$, the existence of a Baire generic set of $C^d$-families of $C^r$-maps $(f_a)_{a\in (-1,1)^k}$ of a manifold $M$ of dimension $\ge 2$, so that for every $a$ small the map $f_a$ has infinitely many sinks. When the dimension of the manifold is greater than $3$, the generic set is formed by families of diffeomorphisms. This result is a counter-example to a conjecture of Pugh and Shub.